Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $n \neq 0$. $a = \dfrac{8n + 6}{-9} \div \dfrac{16n + 12}{7n} $
Dividing by an expression is the same as multiplying by its inverse. $a = \dfrac{8n + 6}{-9} \times \dfrac{7n}{16n + 12} $ When multiplying fractions, we multiply the numerators and the denominators. $a = \dfrac{ (8n + 6) \times 7n } { -9 \times (16n + 12) } $ $ a = \dfrac {7n \times 2(4n + 3)} {-9 \times 4(4n + 3)} $ $ a = \dfrac{14n(4n + 3)}{-36(4n + 3)} $ We can cancel the $4n + 3$ so long as $4n + 3 \neq 0$ Therefore $n \neq -\dfrac{3}{4}$ $a = \dfrac{14n \cancel{(4n + 3})}{-36 \cancel{(4n + 3)}} = -\dfrac{14n}{36} = -\dfrac{7n}{18} $